Numerical and Physical Experiments of Wave Focusingin Short- Crested Seas

Félicien Bonnefoy, Pierre Roux de Reilhac, David Le Touzéand Pierre Ferrant

Ecole Centrale de Nantes, France

Rogue Waves’2004, Brest

Topic of the talk

Generation of deterministic wavepackets in a numerical or physicalwave basin

•Wavemaker motion

•Fully-nonlinear waves

•In 2 and 3 dimensions

(mono- and multidirectional)

Numerical tool:

•Spectral method

•with high-order technique

•Non-periodic

•Specific treatment for wavemaking

Wave elevation record

( )tη

Time in s.

Theoretical Framework

• Potential flow theory

and

• Free surface potential

• Fully-nonlinear free surface conditions

on

unknown: the only nonsurfacic quantity

Time-marchingRunge-Kutta 4

Nonlinear Free surface equations

Time evolution strategy

on

on

Separately approximated by an High-Order technique

Standard Higher-Order Techniques

• The two main methods available are:

– Higher-Order Spectral HOS(West et al 1987, Dommermuth and Yue 1987)Formal and Taylor series expansion of the potential only (not for equations)to obtain the vertical velocity.

– Dirichlet to Neumann Operator DNO(Craig and Sulem 1993)Formal and Taylor series expansion of the DNO only (not for equations) iethe normal velocity.

• Decomposition in recursive Dirichlet problems solved byFourier spectral method and collocation nodes

High-Order Method

• Advantages:– Fast solvers with computational costs in O(NlogN) thanks to the use

of Fast Fourier Transforms.Large number of wave components for random seas or steep wave fields.

– High accuracy of the spectral methods

• Limitations of the HOS method:– Non-breaking cases

Steep wave field involves high order nonlinearities– increase the number of modes– dealiase carefully

– Sawtooth instabilities for very steep wave calculations Standard five-point smoothing applied regularly through the

steepest simulations or decrease of the number of modes

• Standard Higher-Order Simulations– Periodic boundary conditions on the free surface– Initial stage: Free surface elevation and potential specified at t=0– Pneumatic wave generation

F(x+Lx , y) = F(x , y)F(x , y+Ly) = F(x , y)

A High-Order Approach for Wave Basin

• Basin with rigid wallsBy simply changing the basis functions on which we expand our solution:

The natural modes of the basin

• Cosine functionsStill possible to use Fast Fourier Transforms

• A wavemaker to generate the waves starting from rest(no initial wave description required)

• The concept of additionnal potential (Agnon and Bingham 1999)

•Inlet flux conditionsolved in an extended basin

• Resolution of the additional potential in a extended basin

• Extensively validated in a previous second order model(Bonnefoy et al ISOPE’02, Bonnefoy et al OMAE’04)

• In 3D: segmented wavemaker• Improved control laws

(Dalrymple method for large wave angles)

Wavemaker modelling Inlet Flux Condition

Also solved by spectral method

Applications

• Improved deterministic reproduction technique in 2D

• Deterministic reproduction of directional focused wavepackets in 3D

Wave elevation record

( )tη

Time in s.20 30 40

-0.2-0.15

-0.1-0.05

00.05

0.10.15

0.2

Time in s.

Wavemaker motion

( )X t

Deterministic reproduction in 2D

• Wavemaker motion to reproduce this wave field– Control in the frequency domain with a set of components:

amplitude, phase (+ angle in case of 3D generation)

������

���

������

������

�����������������������������������������������������������������������������������������������������������������������������������������

�����������������������������������������������������������������������������������������������������������������������������������

���������������������������������������������������������������������������������������������������������������������������������������������

��

�

z

x

��

��

��

X(t)

Wave probe

X̂(f) C∈

Wavemaker

( )X t

Wave elevation record

( )tη

Time in s.

Characteristics:•Steep wave packet: kpAl = 0.26•Asymetric in time

Basin dimensions: 50m long 5m deep

Analytical methods

• Linear backward propagation: reverse phase method(e.g. Mansard and Funke 1982)

• Second-order bound correction of amplitudes(e.g. Duncan and Drake 1995)

������

���

����

����

����

��������������������������������������������������������������������������������������

��������������������������������������������������������������������������������������

��������������������������������������������������������������������������������������������

��

�

z

x

��

��

�X(t)

Wave probe

Wavemaker

( ) ( ) ( ) ( ) ( ) ( )ˆˆ ˆ ˆikx ikxt f f e X f TF f e X tη η η η→ → → = →-1F F

( ) ( ) ( ) ( )1 2ˆ ˆ ˆt f f fη η η η→ = +F

Wave elevation record

( )tη

Time in s.

20 30 40

-0.2-0.15

-0.1-0.05

00.05

0.10.15

0.2

Time in s.

Wavemaker motion

( )X t

Wavemaker Transfer Function

Iterative corrections

TargetBefore iterationAfter 5 iterations

Initialguess

Non-linearsimulation

Correction on amplitudesand/or phases

Correctedmotion

n=1

ηn (t)Comparison tothe target afterFourier analysis

( )expn n nX X i φ=

( )nX f

Target wave

$ ( )fη

Time in s.

TargetFirst order inputSecond order input

Time in s.

Without iteration With iterations

Elev

ati

on in m

.

Elev

ati

on in m

.

A first step towards higher ordercontrol of nonlinearities

• In litterature: analytical-empirical approach Clauss et al (OMAE’04)

• Crest and trough focusing

• Johannessen et Swan (PRSL 2001)• Zang et al (OMAE 2004)• Bateman (PhD Thesis 2001)

• Separation in odd and even orders

• Phase modification by third order effects is present in odd and evenelevation

( )tη ( )tπη

( )

( )

1212

odd

even

π

π

η η η

η η η

= − = +

( ) ( ) ( )( )

( ) ( )( )

' ' ''

' '

' m n p m n pn n

m n m n

i t k k k xi t k xodd n m n p

n m n p

i t k k xeven m n mn

m n

a e a a a e

a a G e

ω ω ωω

ω ω

η

η

± ± − ± ±− ± ±

> ≥

± − ±± ±

≥

≈ + ≈

∑ ∑

∑

Validation with a small amplitude wave packetSecond order effects

• 10 cm amplitude wave packet (at the focusing point) for 5 m meanwavelength

• Nonlinear effects reduced to second order

• Good agreement between first order and odd elevation,and between second order and even elevation

First orderOdd elevationSecond orderEven elevation

Target wave packetCrest focusingTrough focusing

Measured elevations Odd-even decomposition

Time in s.Time in s.

Third order effects for higher wave amplitude

• Resonant Interactions• No instabilities detected (in contrast with Johannessen and Swan (PRSL 2001)

• Phase velocity

• Non resonant Interactions• Bound terms

• Example with a 30 cm wave packet

Odd and linear elevation Even and second order elevation

linear phase velocitynonlinear phase velocityeven elevation

linear phase velocitynonlinear phase velocityodd elevation

Time in s.Time in s.

Elev

ati

on in m

.

Elev

ati

on in m

.

( )'n ni t k xodd n

na e ωη −=∑

To build the linear elevation

Application to deterministic reproduction

Initial decomposition : linear Initial decomposition : second order

The main features of the focused target wave packet are well reproduced withonly one correction of the wavemaker motion (no iteration so far)

•Central crest and lateral troughs are close to the target both in amplitudes and phases

•Central crest amplitude is correctly estimated

• Better control of the high-frequency waves

Time in s. Time in s.

Elev

ati

on in m

.

Elev

ati

on in m

.

Wavemaker motion corrected with the phase shift

due to nonlinear phase speed modification( ) ( )' iX f X f e φ∆=

Focused wave packet reproduction in 3D

• Directional irregular wave field

S(f,θ) = S(f) D(θ,f)

• Modified Pierson-Moskowitz spectrum (fpeak=0.5Hz, Hs= 4 cm)

• Directional spreading with s=10

• Focusing time t=45 s

• Elevation recorded in 5 locations(probe array used for short-crested seas analysis)

t = 25 s. t = 45 s.

( )5 425 5exp

4 4Sp ps

p

f fHff f f

= −

( ) 2, cos2

D s of θ θθ − =

Reproduction of a directional focused wave field

Analysis in the frequency domain (for the 5 probes)

( )( cos sin ) for 1 to 5i ik x y pa e e FT pφ θ θ η− + = =Three unknowns at each frequency : , ,a φ θ

A set of nonlinear equations solved with a nonlinear least squares method(local minima are expected)

and different initial guesses

1 to 5

random between 0 and 2 gaussian random angle

init p

init

init

a FT

φ πθ

= =

( ) ( ) ( )25

01

T targetp p

pw t t t dtε η η

=

= −∑∫

We obtain a set of solutions of the nonlinear equations: we choose the onethat minimises

Directional focused wave field

Simulated wave packets with the HOS model of the wave basinfor both the focused target and the reproduced wave packet

Target wave fieldfp = 0.5 Hz, Hs = 4 cm

Directional spreading s=10

Focusing time t = 45 s

Reproduced wave field

( ),, , y ta Xφ θ →Prescribed snake-like wavemaker motion

Large waves angles generated with the Dalrymple method

Directional focused wave field

View of the wave field before the focusing event at t = 33.5 s

Target wave field Reproduced wave field

•The main features of the focusing packet are correctly reproduced

•The high-fequency range is underestimated in the predictedwavemaker motion

Directional focused wave field

View of the wave field at the focusing event t = 45 s

Target wave field Reproduced wave field

•Underestimation of the wave crest

•Overestimation of the width of the crest

Conclusion

• High-Order Spectral method applied to the wave generation in awave basin

• Improvement of the wavemaker motion for the generation of deterministic wave packets

• Part of third order effects (phase velocity) taken into account in 2D

• Attempt of deterministic reproduction in 3D

Future work

• Phase velocity correction applied iteratively• Application to different kinds of wave packets (narrow-

banded, broad-banded…)

Comparison between numerical simulations andexperiments

Amplitude 40 cm Amplitude 30 cm

Nonlinear simulation results for linear and second-order wavemaker motion

Difference with the target signal:– Time shift– Peak amplitude– Through amplitude

Improvement of the second ordermodel:

– Reduced time shift

Basin dimensions: 50m long5m deep

Number of modes: Nx=512Nz=64

Order of decomposition: M=5

TargetFirst order inputSecond order input

Time in s.

Analytical methods

Elevation in m.

Wavemaker control

• Successive corrections

• Initial guess: second-order analytical model

Initialguess

Non-linearsimulation

Correction on amplitudesand/or phases

Correctedmotion

n=1

ηn (t)Comparison tothe target afterFourier analysis

( )expn n nX X iφ=

( )nX f

Target wave

$ ( )fη

Numerical checkings

• Number of modes

• Order of decomposition for

• Time interval between successive smoothing (dt = 0.014s)

reference0.45201024

0.08 %0.4516512

2.1 %0.4459256

Relative errorPeak heightNumber of modes

zφ∂∂

reference0. 45165

0.18 %0.45244

0.40 %0.45343

Relative errorPeak heightOrder M

0.09 %0.451930

0.15 %0. 451620

0.02 %0.452240

Reference0.452360

Relative errorPeak heightNumber of timesteps

Iterative corrections1-Phase lag refinement

Wave elevation obtained at iteration « n »

with motion

Correction on the phaseφn+1 = φn + ∆φ with ∆φ = φ − ψn

Simulation result after 5 iterationsTargetBefore iterationAfter 5 iterations

( )expn n nX X i φ=( )exp nn n iη η ψ=

Time in s.

Elevation in m.

Target phase

Iterative corrections2-Amplitude and phase

Correction of the phaseφn+1 = φn + ∆φ with ∆φ = φ − ψn

Correction of the amplitude

with ∆a =

Simulation result after 5 iterationsTargetBefore iterationAfter 5 iterations

1n nX X TF a+ = + ∆

Target amplitude

Elevation in m.

Time in s.

( ) ( )nf fη η−

Iterative Corrections

14163537Error (%)

Iterations onboth the phaseandamplitudes

Iterations onthe phase

First plussecond order

First orderCase

Wave elevation record

( )tη

Time in s.

Comparisoninterval

2

1

2

1

target simu

target

100

T

TT

T

dtError

dt

η η

η

−=

∫∫

Elevation in m.

Time in s.

•Discrepancies for the peak amplitude

•Spurious oscillations before the focused event

Numerical and Physical Experiments of Wave Focusing in Short- Crested SeasTopic of the talkTheoretical FrameworkTime evolution strategyStandard Higher-Order TechniquesHigh-Order MethodA High-Order Approach for Wave BasinWavemaker modelling Inlet Flux ConditionApplicationsDeterministic reproduction in 2DAnalytical methodsIterative correctionsA first step towards higher order control of nonlinearitiesValidation with a small amplitude wave packetSecond order effectsThird order effects for higher wave amplitudeApplication to deterministic reproductionFocused wave packet reproduction in 3DReproduction of a directional focused wave fieldDirectional focused wave fieldDirectional focused wave fieldDirectional focused wave fieldConclusionComparison between numerical simulations and experimentsAnalytical methodsWavemaker controlNumerical checkingsIterative corrections1-Phase lag refinementIterative corrections2-Amplitude and phaseIterative Corrections